SECTION 9.2: ARITHMETIC SEQUENCES and PARTIAL SUMS

[Pages:15](Chapter 9: Discrete Math) 9.11

SECTION 9.2: ARITHMETIC SEQUENCES and PARTIAL SUMS

PART A: WHAT IS AN ARITHMETIC SEQUENCE?

The following appears to be an example of an arithmetic (stress on the "me") sequence:

a1 = 2 a2 = 5 a3 = 8 a4 = 11

We begin with 2. After that, we successively add 3 to obtain the other terms of the sequence.

An arithmetic sequence is determined by:

? Its initial term

Here, it is a1 , although, in other examples, it could be a0 or something else. Here, a1 = 2 .

? Its common difference

This is denoted by d . It is the number that is always added to a previous term to obtain the following term. Here, d = 3.

( ) Observe that: d = a2 - a1 = a3 - a2 = ... = ak+1 - ak k Z+ = ...

The following information completely determines our sequence:

The sequence is arithmetic. (Initial term) a1 = 2 (Common difference) d = 3

(Chapter 9: Discrete Math) 9.12

In general, a recursive definition for an arithmetic sequence that begins with a1 may be given by:

a1 given

( )

ak

+1

=

ak

+

d

k 1; "k is an integer" is implied

Example

The arithmetic sequence 25, 20, 15, 10, ... can be described by:

ad1

= =

25 -5

(Chapter 9: Discrete Math) 9.13

PART B : FORMULA FOR THE GENERAL nth TERM OF AN ARITHMETIC SEQUENCE

Let's begin with a1 and keep adding d until we obtain an expression for an , where n Z+ .

a1 = a1 a2 = a1 + d a3 = a1 + 2d a4 = a1 + 3d

( ) an = a1 + n - 1 d

The general nth term of an arithmetic sequence with initial term a1 and common difference d is given by:

( ) an = a1 + n - 1 d

Think: We take n - 1 steps of size d to get from a1 to an .

Note: Observe that the expression for an is linear in n. This reflects the fact that arithmetic sequences often arise from linear models.

Example

Find the 100th term of the arithmetic sequence: 2, 5, 8, 11, ... (Assume that 2 is the "first term.")

Solution

( ) an = a1 + n - 1 d ( )( ) a100 = 2 + 100 - 1 3

= 2 + (99)(3)

= 299

(Chapter 9: Discrete Math) 9.14

PART C : FORMULA FOR THE nth PARTIAL SUM OF AN ARITHMETIC SEQUENCE

The nth partial sum of an arithmetic sequence with initial term a1 and common difference d is given by:

Sn

=

n

a1

+ 2

an

Think: The (cumulative) sum of the first n terms of an arithmetic sequence is given by the number of terms involved times the average of the first and last terms.

Example Find the 100th partial sum of the arithmetic sequence: 2, 5, 8, 11, ...

Solution

We found in the previous Example that: a100 = 299

Sn

=

n

a1

+ 2

an

( ) S100

=

100

2 + 299 2

( ) =

100

301 2

= 15,050

i.e., 2 + 5 + 8 + ... + 299 = 15,050

This is much easier than doing things brute force on your calculator!

Read the Historical Note on p.628 in Larson for the story of how Gauss quickly

100

computed the sum of the first 100 positive integers, k = 1 + 2 + 3 + ... + 100 . Use our k =1

formula to confirm his result. Gauss's trick is actually used in the proof of our formula; see p.694 in Larson. We will touch on a related question in Section 9.4.

(Chapter 9: Discrete Math) 9.15

SECTION 9.3: GEOMETRIC SEQUENCES, PARTIAL SUMS, and SERIES

PART A: WHAT IS A GEOMETRIC SEQUENCE?

The following appears to be an example of a geometric sequence:

a1 = 2 a2 = 6 a3 = 18 a4 = 54

We begin with 2. After that, we successively multiply by 3 to obtain the other terms of the sequence. Recall that, for an arithmetic sequence, we successively add.

A geometric sequence is determined by: ? Its initial term

Here, it is a1 , although, in other examples, it could be a0 or something else. Here, a1 = 2 .

? Its common ratio

This is denoted by r. It is the number that we always multiply the previous term by to obtain the following term. Here, r = 3.

( ) Observe that: r

=

a2 a1

=

a3 a2

=...=

ak +1 ak

k Z+

=...

The following information completely determines our sequence:

The sequence is geometric.

(Initial term) a1 = 2 (Common ratio) r = 3

(Chapter 9: Discrete Math) 9.16

In general, a recursive definition for a geometric sequence that begins with a1 may be given by:

a1 given

( )

ak

+1

=

ak

r

k 1; "k is an integer" is implied

We assume a1 0 and r 0 .

Example

The geometric sequence 2, 6, 18, 54, ... can be described by:

a1 r

= =

2 3

(Chapter 9: Discrete Math) 9.17

PART B : FORMULA FOR THE GENERAL nth TERM OF A GEOMETRIC SEQUENCE

Let's begin with a1 and keep multiplying by r until we obtain an expression for an , where n Z+ .

a1 = a1 a2 = a1 r a3 = a1 r 2 a4 = a1 r3

an = a1 r n-1

The general nth term of a geometric sequence with initial term a1 and common ratio r is given by:

an = a1 r n-1

Think: As with arithmetic sequences, we take n - 1 steps to get from a1 to an .

Note: Observe that the expression for an is exponential in n. This reflects the fact that geometric sequences often arise from exponential models, for example those involving compound interest or population growth.

(Chapter 9: Discrete Math) 9.18

Example

Find the

6th

term

of

the geometric

sequence:

2,

-1,

1 2

,

...

(Assume that 2 is the "first term.")

Solution

Here,

a1

=

2

and

r

=

-

1 2

.

an = a1 r n-1

( ) a6 =

2

1 6-1 - 2

( ) =

2

-

15 2

( ) =

2

-

1 32

=

-

1 16

Observe that, as n , the terms of this sequence approach 0.

( ) ( ) Assume a1 0 . Then, a1 r n-1 0 as n -1 ................
................

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